Editing Arc measurement

{{Short description|Technique of determining the radius of Earth}}
[[File:Eratosthenes-Erdmessung.svg|thumb|Arc measurement of Eratosthenes]]

'''Arc measurement''',<ref name="Torge2012">{{cite book | last1=Torge | first1=W. | last2=Müller | first2=J. | title=Geodesy | publisher=De Gruyter | series=De Gruyter Textbook | year=2012 | isbn=978-3-11-025000-8 | url=https://books.google.com/books?id=RcfmBQAAQBAJ&pg=PA6 | access-date=2021-05-02 | page=5}}</ref> sometimes called '''degree measurement'''<ref>[[Wilhelm Jordan (geodesist)|Jordan, W.]], & Eggert, O. (1962). Jordan's Handbook of Geodesy, Vol. 1. Zenodo. http://doi.org/10.5281/zenodo.35314</ref> ({{langx|de|Gradmessung}}),<ref name="Torge 2008 p. 5">{{cite book | last=Torge | first=W. | title=Geodäsie | publisher=De Gruyter | series=De Gruyter Lehrbuch | year=2008 | isbn=978-3-11-019817-1 | url=https://books.google.com/books?id=lsO2uz_5euIC&pg=PA5 | language=de | access-date=2021-05-02 | page=5}}</ref> is the [[astrogeodetic]] technique of determining the [[radius of Earth]] and, by [[Circumference#Circle|extension]], [[Earth's circumference|its circumference]]. More specifically, it seeks to determine the local [[Earth radius of curvature]] of the [[figure of the Earth]], by relating the [[latitude]] difference (sometimes also the [[longitude]] difference) and the [[geographic distance]] ([[arc length]]) [[Surveying|surveyed]] between two locations on Earth's surface.<ref>{{Cite book |url=https://books.google.com/books?id=jPVxSDzVRP0C&dq=arc+measurement&pg=PA29 |title=Glossary of the Mapping Sciences |date=1994-01-01 |publisher=[[American Society of Civil Engineers]] (ASCE) |isbn=978-0-7844-7570-6 |language=en}}</ref> The most common variant involves only [[astronomical latitude]]s and the [[meridian arc]] length and is called ''meridian arc measurement''; other variants may involve only [[astronomical longitude]] (''[[parallel of latitude|parallel]] arc measurement'') or both [[geographic coordinates]] (''oblique arc measurement'').<ref name="Torge2012"/>
Arc measurement campaigns in Europe were the precursors to the [[International Association of Geodesy]] (IAG).<ref>{{Cite book|title=IAG 150 Years|volume = 143|last=Torge|first=Wolfgang|date=2015|publisher=Springer, Cham|pages=3–18|language=en|doi=10.1007/1345_2015_42|chapter = From a Regional Project to an International Organization: The "Baeyer-Helmert-Era" of the International Association of Geodesy 1862–1916|series = International Association of Geodesy Symposia|isbn = 978-3-319-24603-1}}</ref>
Nowadays, the method is replaced by worldwide [[geodetic network]]s and by [[satellite geodesy]].

==History==
{{Main|Meridian arc#History}}

The first known arc measurement was performed by [[Eratosthenes' arc measurement|Eratosthenes]] (240 BC) between Alexandria and Syene in what is now Egypt, determining the radius of the Earth with remarkable correctness. 
In the early 8th century, [[Yi Xing]] performed a similar survey.<ref name="Hsu 1993 pp. 90–100">{{cite journal | last=Hsu | first=Mei-Ling | title=The Qin maps: A clue to later Chinese cartographic development | journal=Imago Mundi | publisher=Informa UK Limited | volume=45 | issue=1 | year=1993 | issn=0308-5694 | doi=10.1080/03085699308592766 | pages=90–100}}</ref>

The French physician [[Jean Fernel]] measured the arc in 1528. The Dutch geodesist [[Willebrord Snellius|Snellius]] (~1620) repeated the experiment between [[Alkmaar]] and [[Bergen op Zoom]] using more modern geodetic instrumentation (''[[Snellius' triangulation]]'').

Later arc measurements aimed at determining the [[flattening]] of the Earth ellipsoid by measuring at different [[geographic latitude]]s. The first of these was the ''[[French Geodesic Mission]]'', commissioned by the [[French Academy of Sciences]] in 1735–1738, involving measurement expeditions to Lapland ([[Pierre Louis Maupertuis|Maupertuis]] et al.) and Peru ([[Pierre Bouguer]] et al.).

[[Friedrich Georg Wilhelm von Struve|Struve]] measured a [[geodetic control network]] via [[triangulation (surveying)|triangulation]] between the [[Arctic Sea]] and the [[Black Sea]], the ''[[Struve Geodetic Arc]]''. 

[[File:'The Adventures of Three Englishmen and Three Russians in South Africa' by Jules Férat 15.jpg|thumb|Imaginary arc measurement described by [[Jules Verne]] in his book ''[[The Adventures of Three Englishmen and Three Russians in South Africa]]'' (1872).]]

=== Chronological list ===
This is a partial chronological list of arc measurements:<ref>{{cite book | title=A history of the determination of the figure of the earth from arc measurements |author= Butterfield, Arthur Dexter | publisher=Internet Archive | date=1906 | page=156 | url=https://archive.org/details/ahistorydetermi00buttgoog/page/n168/mode/1up | ref={{sfnref|Internet Archive|2016}} | access-date=2025-01-26}}</ref><ref>{{cite web | title=Arc Measures | website=International Institution for the History of Surveying and Measurement | url=https://www.fig.net/organisation/perm/hsm/history_of/measurements/arc.asp | ref={{sfnref|International Institution for the History of Surveying and Measurement}} | access-date=2025-01-27}}</ref>
* 230 B.C.: [[Eratosthenes' arc measurement]]
* 100 B.C.: [[Posidonius' arc measurement]]
* 724 AD: [[Yi Xing's arc measurement]]
* 827 A.D.: [[Al-Ma'mun's arc measurement]]
* 1528: [[Fernel's arc measurement]]
* 1617: [[Snellius' survey]]
* 1633-1635: [[Norwood's arc measurement]]
* 1658: [[Riccioli and Grimaldi's arc measurement]]
* 1669: [[Picard's arc measurement]]
* 1684-1718: [[Dunkirk-Collioure arc measurement (Cassini, Cassini, and de La Hire)]]
* 1736-1737: [[French Geodesic Mission to Lapland]] 
* 1735-1739: [[French Geodesic Mission to the Equator]]
* 1740: [[Dunkirk-Collioure arc measurement (Cassini de Thury and de Lacaille)]]
* 1750-1751: [[Maire and Boscovich's arc measurement]]
* 1752: [[De Lacaille's arc measurement]]
* 1791-1853: [[Principal Triangulation of Great Britain]]
* 1792-1798: [[meridian arc of Delambre and Méchain]]
* 1802–1841: the [[Great Trigonometric Survey of India]]
* 1806-1809: [[Arago and Biot's arc measurement]]
* 1816-1855: [[Struve Geodetic Arc]]
* 1821-1825: [[Gauss' geodetic survey]]
* 1841-1848: [[Maclear's arc measurement]]
* 1879: [[West Europe-Africa Meridian-arc]]
* 1899-1902: [[Swedish–Russian Arc-of-Meridian Expedition]]
* 1921: [[Hopfner's arc measurement]]

==Determination==
Assume the [[astronomic latitude]]s of two endpoints, <math>\phi_s</math> (standpoint) and <math>\phi_f</math> (forepoint) are known; these can be [[latitude determination|determined]] by [[astrogeodesy]], observing the [[zenith distance]]s of sufficient numbers of [[star]]s ([[meridian altitude]] method). 

Then, the empirical [[Earth's meridional radius of curvature]] at the midpoint of the meridian arc can then be determined inverting the [[great-circle distance]] (or [[circular arc length]]) formula:
:<math>R = \frac{\mathit{\Delta}'}{\vert\phi_s - \phi_f\vert}</math>
where the latitudes are in radians and <math>\mathit{\Delta}'</math> is the [[arc length]] on [[mean sea level]] (MSL).

Historically, the distance between two places has been determined at low precision by [[Pacing (surveying)|pacing]] or [[odometry]].

High precision land surveys can be used to determine the distance between two places at nearly the same longitude by measuring a [[baseline (surveying)|baseline]] and a [[triangulation network]] linking [[Benchmark (surveying)|fixed points]]. The [[meridian distance]] <math>\mathit{\Delta}</math> from one end point to a fictitious point at the same latitude as the second end point is then calculated by trigonometry. The surface distance <math>\mathit{\Delta}</math> is reduced to the corresponding distance at MSL, <math>\mathit{\Delta}'</math> (see: [[Geographical distance#Altitude correction]]).

==Extensions==
[[File:Clarke figure of the Earth 1860 (cropped).jpg|thumb|upright=2|right|Arc measurements used in Clarke's Figure of the Earth, 1858]]
Additional arc measurements, at different latitudinal bands (each delimited by a new pair of standpoint and forepoint), serve to [[Earth ellipsoid#Determination|determine Earth's flattening]].
[[Friedrich Bessel|Bessel]] compiled several [[meridian arc]]s, to compute the famous [[Bessel ellipsoid]] (1841).
[[Alexander Ross Clarke#Figure of the Earth, 1858|Clarke]] (1858) combined most of the arc measurements then available to define a new [[reference ellipsoid]].<ref>{{cite EB1911
| ref = {{harvid|Britannica|1911a}}
| last  = Clarke
| first = Alexander Ross
| last2  = Helmert
| first2 = Friedrich Robert
| author2-link  = Friedrich Robert Helmert 
| wstitle = Earth, Figure of the
| volume = 8
| pages  = 801–814
}}</ref>

== See also ==
* [[Astrogeodesy]]
* {{ill|Central European Arc Measurement|de|Gradmessungskommission}}
* [[Earth ellipsoid]]
* [[Geodesy]]
* {{section link|Gradian|Relation to the metre}}
* [[History of geodesy]]
** {{section link|Spherical Earth|History}}
** {{section link|Meridian arc|History}}
** {{section link|Earth's circumference|History}}
* [[Meridian arc]]
* [[Paris Meridian]]

==References==
{{Reflist}}

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{{DEFAULTSORT:Grade Measurement}}
[[Category:Geodetic surveys]]
[[Category:History of measurement]]